Physics Problem: Hydrogen Gas In A Sphere

Hey guys, let's dive into a cool physics problem today involving hydrogen gas inside a sphere. We've got a sphere with a radius of 0.1 meters, and inside it, hydrogen is hanging out at a cozy temperature of 27 degrees Celsius. The real kicker here is that the mean free path of the hydrogen molecules is exactly the same as the diameter of the sphere. Our mission, should we choose to accept it, is to figure out the pressure of the hydrogen and the concentration of its molecules. Don't worry, we'll break it down step-by-step so it's super clear. This kind of problem is awesome for understanding the behavior of gases under specific conditions.

Understanding the Concepts

Before we crunch the numbers, let's get our heads around some key physics concepts. We're dealing with gases, so the Ideal Gas Law is going to be our best friend. It relates pressure (P), volume (V), the number of moles (n), the ideal gas constant (R), and temperature (T) with the formula PV = nRT. We'll also need to think about molecular concentration, which is basically the number of molecules per unit volume. And of course, the mean free path is super important here – it's the average distance a molecule travels before it collides with another molecule. The problem gives us a direct link between this mean free path and the size of our sphere, which is a crucial piece of information.

To tackle this, we'll need to convert our temperature from Celsius to Kelvin, because physics loves Kelvin! So, 27°C becomes 27 + 273.15 = 300.15 K. For simplicity in calculations, we often use 300 K. The radius of the sphere is 0.1 m, which means its diameter is 0.2 m. The problem states that the mean free path ($\lambda$) is equal to the diameter of the sphere, so $\lambda = 0.2$ m. This is quite a large mean free path relative to the size of the container, which tells us something interesting about the gas conditions – it's likely a low-pressure environment. We'll also need the Boltzmann constant ($k_B$), which is approximately $1.38 \times 10^{-23}$ J/K, and the molar mass of hydrogen ($M$), which is about 0.002 kg/mol.

The Ideal Gas Law and Molecular Concentration

Now, let's get down to business with the Ideal Gas Law. We can rewrite it in terms of the number of molecules (N) and the volume (V) using the relationship $n = N/N_A$, where $N_A$ is Avogadro's number (approx. $6.022 \times 10^{23}$ mol$^{-1}$). So, $PV = (N/N_A)RT$. We also know that $R = N_A k_B$. Substituting this in, we get $PV = N k_B T$. This form is super handy because it directly relates pressure, the number of molecules, and temperature.

We're looking for molecular concentration, which is $n = N/V$. From the modified Ideal Gas Law, we can rearrange to find $N/V$: $P/ (k_B T) = N/V$. So, molecular concentration ($n$) = $P / (k_B T)$. This equation tells us that concentration is directly proportional to pressure and inversely proportional to temperature. Higher pressure or lower temperature means more molecules packed into the same space. This is a fundamental relationship in understanding gas behavior and will be key to solving our problem. Remember, $k_B$ is the Boltzmann constant and $T$ is the absolute temperature in Kelvin.

The Mean Free Path Formula

To link the given mean free path to our problem, we need its formula. The mean free path ($\lambda$) for a gas is given by: $\lambda = \frac{1}{\sqrt{2} \pi d^2 n}$, where $d$ is the diameter of a gas molecule and $n$ is the molecular concentration. This formula is derived from kinetic theory and shows that the mean free path is inversely proportional to the square of the molecular diameter and the molecular concentration. A larger molecule or a higher concentration of molecules will lead to shorter mean free paths because there are more obstacles for a molecule to run into. Conversely, a lower concentration or smaller molecules mean molecules can travel further on average between collisions. This is a critical insight for our problem, as the relationship between $\lambda$ and $n$ will allow us to solve for the unknown concentration.

We are given that $\lambda$ is equal to the diameter of the sphere. Let's denote the diameter of the sphere as $D_{sphere}$. So, $\lambda = D_{sphere}$. We calculated $D_{sphere} = 0.2$ m. The formula for the mean free path involves the molecular concentration ($n$). We can rearrange the mean free path formula to solve for $n$: $n = \frac{1}{\sqrt{2} \pi d^2 \lambda}$. Here, $d$ represents the diameter of a single hydrogen molecule. This is a piece of information we don't explicitly have, but we can often estimate it or it might be implicitly defined by other properties. However, there's a more direct way to use the given information. The problem states the mean free path is the diameter of the sphere. This implies a specific density of molecules within that sphere. We can use the mean free path formula to find the molecular concentration, $n$, if we knew the molecular diameter, $d$. But wait, the problem gives us a direct relationship: $\lambda = D_{sphere}$. This means we can use the mean free path formula to determine the molecular concentration ($n$) by substituting the given $\lambda$ value. The formula is $\lambda = \frac{1}{\sqrt{2} \pi d^2 n}$. We are given $\lambda = 0.2$ m. We need $d$, the molecular diameter of hydrogen. This is where it gets a bit tricky if $d$ isn't provided. However, let's look at the relationship between $\lambda$ and $n$ again: $\lambda = 1 / (\sqrt{2} \pi d^2 n)$. We can rearrange this to solve for $n$: $n = 1 / (\sqrt{2} \pi d^2 \lambda)$. We need $d$. What if we don't need $d$? Let's rethink.

The problem statement is quite specific: